Method of forming gears



Filed May 4, 1928 3 Sheets-Sheet 1 INVENTOR Ell/Mt 'March 24, 1931.

E. WILDHABER I 1,797,461

METHOD OF FORMING GEARS Filed May 4, 1928 3 Sheets-Sheet 2- INVENTOR March 24, 1931. E. WILDHABER 1,797,461

METHOD OF FORMING GEARS Filed May 4, 1928 3 Sheets-Sheet 5 Fl G18 m ,1: ,18 mm INVENTOR R E 1 0 94 E/Lww t- Patented Mar. 24, 1931 UNITED STATES ERNEST WILDHABER, OF BROOKLYN, NEW YORK METHOD OF FORMING GEARS Application filed May 4,

The present invention relates to methods of forming gears, and particularly of gears having differing profiles in parallel planes perpendicular to their axes. The present invention is intended especially for use'on gears having angularly disposed axes, such as worms and worm gears, bevel gears and hypoid gears, but its principles can also be ap lied to gears with parallel axes.

ne object of the present invention is to provide a novel method for accurately and expeditiously forming a pair of gears at least one of which has difi'ering tooth profiles in parallel planes perpendicular to its axis. Another object is to provide a novel method for accurately and efiiciently forming gear pairs of novel tooth form, either by cutting or by grinding.

Another aim is to provide a method for accurately and efiiciently cutting a novel pair in which the worm contains differing thread profiles in parallel planes perpendicular to its axis, and a method for grinding both members of said pair of worm of worm gears,

gears, worm and worm wheel.

Difliculty has hitherto often been experienced in cutting a pair of worm gears of large size and comparatively small ratio, inasmuch as a hob of the size of a large worm is an expensive tool, especially when accurately made. The present invention also aims to remedy this condition, by providing a method that shall permit to cut said pair of worin gears with hobs of standard size. Moreover a method shall be devised which permits to cut one of a air of Worm gears in a bobbing op eration, m which the hob contains a smaller number of threads than the gear which is to mesh with the finished gear blank, and which 0 can be carried out on existing machines.

Further a method of eiiiciently cutting a pair of mating worm gears shall be devised, of which both gears of said pair, frequently called the worm and the wheel, contain differing profiles in parallel planes perpendicu lar to their respective axes.

Another chief object of the present invention is to introduce a novel principle to the art of gear forming, and to apply this principle to methods of forming all kinds of Serial No. 275,143.

gears, and particularly-to methods of forming gears with angularly disposed axes, such as Worm gears, hypoid gears and bevel gears.

A further object is to provide a method of forming a pair of gears by providing a pair of forming members containing formingsurfaces of diverging longitudinal profile, as will be further explained hereafter, and by forming said pair of gears with said pair of forming members, by rotating a forming member in engagement with a gear blank, by turning the gear blank on its axis, and by providing feeding motion between forming member and gear blank in a manner to approach the axes of the forming member and ot the gear blank while maintaining said axes in constant angular relation.

Another object is to provide a method of forming a pair of gears having angularly disposed axes, by providing a pair of hobs having cutting edges disposed in single threads of diverging longitudinal profile, the profile of axial sections of the threads of said hobs being substantially complementary and being frequently made straight, ar il by cutting said pair of gears with said pair of hobs, by rotating a hob and a gear in timed relation, and by providing feeding motion to effect a final relative position between hob and gear blank, in which the hob may finish the whole tooth surfaces or thread surfaces of the gear blank.

A still other object is to provide a method of efficiently forming a pairof gears having angularly disposed axes, by forming each gear of said pair with a hob having cutting teeth arranged in a thread, by rotating said hob and a gear blank, and by providing feeding motion in a manner to first rough out the gear blank and then to finish the gear blank in a single position of feed. Another aim is to provide a method of this said character, in

which said feed motion is efi'ected at a changing rate, to obtain a coarse a fine finishing cut in the eration.

A number of other objects will appear the course of the specification and from recital of the appended claims. q

roughing cut and same cycle of op- My invention will be explained by Way of examples illustrated in the accompanying drawings, in which:

Fig. 1 is a diagrammatic view of a pair of worm gears such as may be formed in accordance with the present invention, the view being taken in the direction of the axis of the larger member of the pair, frequently called worm wheel.

Fig. 2 is a view and partly a section along lines 1-1 of Fig. 1.

Fig. 3 is a view of a worm gear in enagement with one species of forming memer, illustrative of one embodiment of the present invention.

Fig. 1 is a partial normal section through the teeth or threads of the worm gear shown in Fig. 3, alon lines 2--2 of Fig. 3, and an axial section 0' the forming member shown in Fig. 3. For convenience, the tooth section isshown longer than corresponds to the width of face of gear 12.

Fig. 4a and Fig. 4b are auxiliary views illustrative of forming members having'm'odified surfaces.

Fig. 5 is a view of a worm gear in engagement with another species of forming member, frequently embodied asa hob, and illustrative of the present invention.

Fig. 6 is a partial normal section through the teeth or threads of the worm gear shown in Fig. 5, and an axial section of the forming member or hob shown in Fig. 5.

Fig. 7 is an end view of a helicoidal thread and a diagram illustrative of a computation carried out for use in methods according to the present invention.

Fig. 8 and Fig. 9 are diagrammatic plan views of pairs of worm one gear of the pair'contains differing profiles in parallel planes perpendicular to its axis. Fig. 9 is furthermore a view explanatory of a certain manner of computation hereafter referred to.

Fig. 10 is a diagrammatic plan view of a device for bobbing gears-while providing a feeding motion at a changing rate, in accordance with the present invention.

Fig. 11 is a normal section through the threads of a large worm having differing profiles in parallel planes perpendicular to its axis, and an axial section through a forming member. Fig. 11 can also be considered as a normal section through a worm wheel. Fig. 11 illustrates in either case the use of a different pressure angle for generating as compared with the pressure angle of the finished pair of gears.

Fig. 12 is a section similar to Fig. 11, and illustrative of a forming member of hob shape. i

Fig. 13 and Fig. 14 are normal sections through the threa s or teeth of gear blanks and through threads of two modified hobs, to be used in accordance with the present invention.

gears, in which only Fig. 15 is a perspective diagram illustrative of a mean line of action between a pair of worm gears, and of certain relations between the mesh of said worm gears and a plane perpendicular to said line of action, adjacent a mean oint of mesh.

Fig. 16, Fig. 1%, Fig. 18 and Fig. 19 are diagrams explanatory of certain interrelations observed in establishing the proportions and setting of a forming member in accordance with the present invention. The said diagrams are in the nature of topographic maps of the tooth surfaces adjacent a mean point of the line of action.

Fig. 20 is a diagrammatic plan view of a fornnngmember and a tapered gear blank, illustrative of a modification of the present invention.

Fig. 21 is a diagrammatic plan view of a forming member or hob and a spiral bevel gear blank, illustrative of a further embodiment of my invention.

Fig. 22 is a diagrammatic and partial plan view of a tapered forming member or hob and of a spiral bevel gear blank suited to mesh with the spiral bevel gear referred to in Fig. 21.

Fig. 23 is an axial section of the pair of forming members or hobs diagrammatically indicated in the Figures 21 and 22.

In the Figures 1 and 2 the numerals 11 and 12 denote a pair of worm gears of comparatively small ratio, rotatable on axes 13, 14 which are angularly disposed to each other, namely disposed at right angles in the illustrated instance. Gearing of this character has hitherto often been embodied by a pair of gears having helical teeth or threads of constant profile, both gears being conjugate to the same rack. The gears then could be out like ordinary helical gears with standard tools, and present no difiiculty or additional ex ense.

on both gears (11 and 12) are provided with helical teeth, they are, however, notfully conjugate to each other, as is known. They are suited to transmit uniform motion, but contact with each other only in a point or in points at a time, whereas fully conjugate gears contact with each other along lines.

It is possible to cut the gears (11 and 12) with existing methods in a manner to effect mesh with line contact, by using the known proceduresof forming a worm and a conjugate wheel, and by cutting the worm wheel (12) with a hob corresponding to the worm. When the size of the worm is large, the hob embodying the worm is also large and thereby presents a serious drawback. Other known methods necessitate the use of special machines and often present other complications.

1 shall now proceed to describe a novel method, with which the gears 11 and 12 may be formed to mesh either with line contact or with any desirable approximation of line contact, and which permits the use of standard tools, which is very efiicient, and which can be carried out on existing machines, if so desired. When the gears 11 and 12 are provided both with helical teeth with a known method, they mesh with point contact, as already mentioned, and a point of contact moves during the mesh along a line of action 15 or 16. These lines are straight and perpendicular to the tooth surfaces when the gears 11 and 12 are provided with involute tooth forms, that is to say with the tooth forms now in general use. The involute form of teeth, as well known, may be derived from a basic rack which contains plane tooth sides, and may be produced in conventional man- .ner with hobs having substantially straight cuttin ed es.

Ordinarily the two lines of action 15, 16 intersect in a point (17), which is frequently called the pitch point and in which the axial pitch of gear 11 equals the circular pitch of gear 12.

Instead of forming both gears 11 and 12 conjugate to a rack and thereby obtaining point contact between the two gears, I form at least one of the gears conJugate to a surface which may mesh with said gear also along a line of action 15, or 16, but whlch 1s conxev in a direction lengthwise of the teeth. According to one embodiment of the invention, a milling cutter or grinding wheel 20 is provided, see Fig. 3 and Fig. 4, and set so that its axis 21 intersects the normal or line of action 15, and in the view Fig. 3 coincides with the projected normal 15. Forming member 20 contains a straight profile 22 in an axial section, see Fig. 4, and lts forming surface is a conical surface. When this surface is moved in the direction of its axis 21, it remains per endicular to line 15 as are also the contacting tooth surfaces of the gears 11 and 12 immediately ad]acent a point of contact along line 15.

The mesh between the two gears 11 and 12 along line 15 takes place at a uniform rate, that is to say a point of contact moves along the line of action 15 exactly in proportion to the turning angles of the gears, and it moves a distance, which will be called perpendicular itch, per motion of the gears by one toot Forming member 20 can therefore be made to mesh along line 15 with a gear blank, when itis moved along its axis 21 at such a uniform rate, that its intersection point with line 15 also moves one perpendicular pitch per angular motion of the gear blank by one tooth. In the position of the forming member shown in full lines, it contacts at a point 24 of line 15 with gear blank 12. A further position of the forming member 20 in its path alon its axis 21 is indicated in dotted lines 20 in ig. 3.

The conical surface of the forming member 20 naturally removes less stock from the blank, than a surface side of a rack tooth. The amount of stock left on the sides of line 15 depends on the diameter of the conical forming wheel, a wheel of larger diameter removing more stock than a wheel of smaller diameter. The conical surface of a forming wheel of very ameter would practically be identical with a plane tooth side of a rack, in the zone of mesh, and therefore remove too much stock on the sides, effecting point contact between mating gears. By suitably selecting the diameter of the forming wheel, stock may be removed insuch a measure that line contact between the mating gears results, or a contact which approximates line contact and yet allows slight misalignment.

novel and practical method for computing the size of the forming wheel will be described hereafter. It forms an important part of the present invention, inasmuch as the complete success of the method depends largely on the proper selection of a forming wheel, and inasmuch no such or similar mannor of computation existed, as far as I am aware, either in this or in other fields.

The degree ofdivergence of the conical surface of the forming member from the plane side of a rack depends not only on the diameter of the forming wheel, but also on the inclination angle a of its profile 22 with respect to aplane perpendicular to its axis. The de ree of divergence, and with it the degree of removing stock outside of line 15, can be measured with the radius of curvature of the forming surface in a plane containing normal 15 and extending in the direction of the circumference, that is to say in longitudinal direction. In other words the degree of divergence can be measured with the radius of curvature of the forming surface in a plane perpendicular to the plane of the drawing Fig. 4 and containing line 15. It can be demonstrated with the known means of mathematics, that this radius of curvature equals distance 24-25, point 25 being the intersection point between normal 15 and axis 21, and being the center of said curvature. It is this quantity 24-25 which will be especially computed hereafter.

Instead of providing a forming member embodying the plane large di' 20 of conical form, a forming member 26 may by a hob, and is then provided with cutting teeth 28 indicated in dotted lines.

Forming member 26 is then an involute hob of conventional character, and is suited to mesh along two lines of action 15, 16 when set in customary manner to its lead angle relatively to the projection of said lines 15. 16, Fig. 5. The forming surface of thread 27 has a character similar to the conical surface of member 20, with respect to its divergence from the plane side of a rack. Like said conical surface it contacts with the plane side of said rack along a straight line, and adjacent said straight line it can be very closely approximated by a conical surface having an axis parallel to the axis of the hob, as is known to those familiar with gear mat-hematics. In the case of a conical surface (20) the center of curvature 25 referred to lies on the intersection between normal 15 and axis 21. In the present case normal 15 does not intersect the axis 3() of thread 27 and the corresponding center 31 of curvature lies on normal 15 at the point, where said normal comes closest to axis 30 of the thread. This point is also the point of tangency between surface normal 15 (see Fig. 7) and the cylindrical base surface 32 of the involute helicoidal thread. The actual length of radius of curvature 24-31 may be computed with the following formula which can be derived with the known methods of mathematics:

sin a "1 cos (1- cos it Herein R. is the considered radius of the thread 27, from center 30 to point 24, h denotes the lead angle at said radius, and a is the pressure angle or inclination angle of line 15 with respect to the plane of the drawing Fig. 5. Angle a is clearly seen in Fig. (5. Angle a is also the pressure angle of a rack conjugate to the thread 27 and having teeth extending in a direction inclined by an angle h to a plane perpendicular to the axis 30 of thread 27.

The present invention can be practiced for forming gear pairs as indicated in the Figures 8 and 9 as Well as for forming gear pairs according to Fig. 1 and Fig. 2. According to Fig. 8 the worm 34 or smaller gear of a pair of worm gears 34, 35 contains differing profiles in parallel planes perpendicular to its axis 36, while the gear 35 is provided with helical teeth of constant profile. Gear 35 may therefore be produced according to any suit-- able known method, while only worm 34 is formed with a method according to the present invention. According to Fig. 9 the worm 34 contains helical threads, and may be formed in any suitable known manner, and only the worm gear 35 contains differing profiles in parallel planes perpendicular to its axis 37. In this case gear 35 has the character of a known worm wheel.

Fig. 9 also illustrates a somewhat modified form of mesh, inasmuch as the lines of action 15', 16' do not intersect, but are offset from one another. Intersection is not a necessity, between the two lines of action (15, 15) which form the basis of computation and settings in accordance with the present invention. The only requirement for the two lines of action 15, 16' is a position such that a force acting in the direction of a line of action (15' or 16') exerts turning moments on the two gears of a pair in proportion to the respective numbers of teeth or threads.

In the case illustrated in Fig. 3 and Fig. 4, the gear blank is periodically indexed, namely after finishing one or both sides of a. tooth space. Feed between forming member 20 and gear blank 12 is in the direction of axis 21, at an angle to the direction of the axis 14 of the gear blank.

In the case illustrated in Fig. 5 and Fig. 6, the gear blank is continuously indexed, while the hob or forming member 26 is rotated in timed relation to the gear blank. Feeding motion between forming member 26 and gear blank 12 is effected preferably in a direction at right angles to the axis 14 of the gear blank, that is to say along a radius of the gear blank. The axes of the hob and of the ear blank are thereby gradually approached until full cutting depth is reached. The feed is then stopped or reversed. Before reaching full cutting depth the hob does not mesh with the final tooth surfaces or thread surfaces of the gear blank, and therefore applies only a roughing cut. The finishing cut is applied immediately before reaching full cutting depth and at full cutting depth. While roughing and finishing may form part of the same single cycle of operations, the rough cutting operation and the finish cutting operation arenevertheless separated in time. hob does either roughing or finishing work, but does not perform both operations simultaneously. This feature makes it possible to combine a heavy and efficient roughing cut with a fine finishing out, which furnishes smoothly finished tooth surfaces as well as high accuracy In accordance with the present invention the feeding motion between the gear blank and the hob is effected at a changing rate, in a manner to decrease with increasing depth of cut.

A device for effecting a changing rate of feed is diagrammatically indicated in Fig. 10. A hob 38 and a gear blank 40 are rotated on their axes in timed relation to each other, the shafts of the hob and the gear blank being operatively connected in a known manner not indicated in the drawing. Gear blank 40 is mounted on a slide 41., which may be fed towards hob 38 in direction of arrow 42 by means of a cam 43. Cam 43 is mounted on slide 41, and is geared up with the gear blank quire different radii of by means of gears 44 indicated through their pitch circles only. Cam 43 is geared 1n a manner to perform one revolution per cycle of operations, that is to say during the t 1me consumed by roughing, finishing, removing, and replacing a gear blank. The shown cam acts on a stationary projection 45 and thereby efiects positive motion in a direction towards the gear blank. The return motion of slide 41 may be effected by another cam or by a strong spring, not shown in the drawing.

The Figures 11 and 12 illustrate a refinement of my invention as so far described. It has already been pointed out that the cutting action of a forming member depends not only on its diameter, but on itspressure angle as well, in such a manner that the radius of curvature in longitudinal direction is a measure of the cuttin action. It is therefore possible to obtain d ifi'erent cutting actions with forming members of standard diameters, by providing different pressure angles a of the forming member. Moreover a gear pair may be adapted to a given forming member by changing the pressure angle of its lines of action 15, 16. Different pressure angles a recurvature R and the pressure angle of a gear pair can usually be so selected as to suit an existing forming member or hob.

In all such cases a gear blank 11 is made to mesh with a forming member 45 or a forming member 46 of hob form along a line of action 15 (or 16) which has a diiferent inclination or pressure angle a than the line of action 15, along which the completed gear pair meshes. And the inclination or pressure angle a of the forming member (45 or 46) is different from the pressure angle a of line of action 15, that is to say from the angle a at which line 15 is inclined to a plane parallel to the two axes of the pair of gears.

Angle a of the forming member may be made larger or smaller depending on whether a small or a large radius R of curvature is required, as determined by the computation hereafter described. In the instance illustrated in Fig. 11 and Fig. 12, the pressure angle a is smaller than the pressure angle a of the finished gear pair; and the curvature radius R of the forming tool is larger than pressure angle a equal to the pressure angle of the gear blank.

he center of curvature is again located either at the point of intersection 47 between normal 15 and axis 48 of forming member 45 (Fig. 11) or at the point 49 (Fig. 12) at which normal 15comes closest to the axis 50 of forming member 46 (Fig. 12). In the latter figure, the axis of the conical surface of curvature of the involute helicoidal surface of thread 51 is indicated in dotted lines 52, and the profile of said conical surface is indicated in dotted lines 53.

In some cases, as for instance when a worm gear meshing with a multiple thread worm is to be produced with a single thread hob set at an acute angle to the axis of the gear blank, a hob of cylindrical outline would cut grooves or spaces having a radius in lengthwise direction larger than required to accommodate a cylindrical worm, which is set at right angles to the axis of the finished gear. In such and other cases a hob 54 of convex outline may be provided, see Fig. 13, without changing the shape of the thread 55 itself. The thread may be made of involute helicoidal form, as in the examples heretofore described.

In other cases it is desirable to combine the principles of the present invention with those disclosed in my copending application Serial N 0. 275,142 filed on even date herewith, namely May 4, 1928. In such cases a hob 56 may be provided, see Fig. 14, having cutting edges 57 disposed at a changing inclination with respect to the axis 58 of the hob.

oreover the present invention is not conned to gearing with straight lines of action 15, but is generally applicable to all kinds of gearing. In all cases of worm gear pairs, one may base the computation on the line of action which exists between a pair of gears having helical teeth. This line of action is a straight line 15 when such helical teeth contain-involute helicoidal tooth sides. tooth forms generally furnish of action,

members may be provided with files, as indicated at 59 and 60 in 4a and 4b.,-

Computation will now be explained, referring first to Fig. 15. In this figure, 61 and 62 denote a pair of worm gears, indicated in curved prothe Figures position, in which it intersects line of action 15 in a point 69. This point is frequently made also the contact point between auxiliary pitch surfaces (61', 62) of cylindrical shape, and is here so assumed. Let r and r" be the distances of point 69 from the axes 63 and 66, or the radii of the said auxiliary pitch plane parallel to the axes 63 and 66;

-. of which lines 70 and 71 and which at point 69 have surfaces'61, 62. Let again a: denote the pressure angle of line of action 15, that is to say the angle included between line 15 and a and let it be the angle between the direction of the rack teeth and a plane perpendicular to axis 63. When point 69 is the contact point between said auxiliary pitch surfaces, then angle h can also be considered as the lead angle of worm gear 61 at the radius r.

We will first consider the mesh between imagined involute helicoidal tooth surfaces. Such surfaces mesh with point contact as known. It will then be shown, how the deee of said point contact or its deviation rom line contact may be computed, in accordance with the resent invention, and how its deficiency can e remedied. As already pointed out a remedy is effected by forming at least one of the gears so that it contains nonhelical teeth, that is to say differing tooth profiles or thread profiles in parallel planes at right angles to its axis.

In considering the mesh of the imagined involute helicoidal tooth surfaces, such surfaces of worm gear 61 mesh with plane 68 in a manner to contact with it along a straight line 70, which is parallel to the projection of axis 63 into plane 68, as is known to those familiar with gear mathematics. Moreover the imagined involute helicoidal teeth of worm gear 62 mesh with plane 68 in a manner to contact with it along a straight line 71, which is parallel to the projection of axis 66 to plane 68. The pitch line of the rack is indicated as a dash and dot line 72 which is the intersection of the pitch plane of the rack with the plane tooth sides 68. The said pitch plane, as known, is parallel to the two axes 63, 66.

The nature of the helical tooth surfaces will now be further analyzed adjacent their point of contact 69. Fig. 16 shows the plane side 68 of the rack laid down in the plane of the drawing. The imagined helical tooth surfaces of the gears 61, 62 contact with the plane (68) of the drawing along the afore said lines 7 O, 71. With a high degree of approximation, the helical tooth surfaces themselves can be considered as conical surfaces, are generatrices, curvature radii C, in a normal section perpendicular to lines 70, 71 respectively. A curvature radius C is determined in the manner described for curvature radius R, and equals the actual length of distance 69-65 and of distance 69-67 respectively, see Fig. 15.

Instead of substituting a conical surface in place of an involute helicoidal surface, a cylindrical surface of the same curvature radius (lmay be substituted, with the same close degree of approximation, when only parts immediately adjacent point 69 are considered. A surface of this character is known as a curvature surface in mathematics. In Fig. 16 the lines 71 drawn parallel to contact line 71 are generatrices of one of said cylindrical curvature surfaces, having each a constant distance from the plane (68) of the drawing along their whole length. The various indicated lines 71' correspond to distances which increase in the manner of an arithmetic progression, namely as 1, 2, 3, 4. The lines 71' form a topographical map of the considered tooth surface adjacent point 69, by constituting lines of equal altitude or elevation from the plane of the drawing. The lines 71a correspond to a certain altitude from the plane of the drawing; the lines 71'?) correspond to double said altitude and so on.

In a similar way, lines 70 drawn parallel to line of contact 70 are lines of equal altitude or elevation, and form a topographical ma of the other considered tooth surface, whic contacts with the plane (68) of the drawing along line 70. Lines 70' correspond to distances on the opposite side of plane 68 as compared with the distances of line 71'. In other words the tooth surfaces contact with the plane of thedrawing (68) from op osite sides. For thisreason lines 70 are s own dotted.

In the considered area, the distance of any point of a tooth surface or of its cylindrical curvature surface is proportional to the square of its direct distance 7 from the line of contact 70 or 71. The distance (2,) of any such point from the plane of the drawing equals 1 ,y "2 in? so when y and 3 denote the direct or normal distance of a considered point from the lines 70 and 71 respectively, and when G and C denote the respective radii of curvature. This relation is well known in mathematics it is the distance of a point of a circle or parabola from its tangent.

Considering any point, such as point 74, it will have a distance a from the plane of the drawing, when point 74 is a point of the tooth surface extending along line 70. And it will have a distance 2 from the plane of the drawing, when point 7 a is a point of the tooth surface extending along line 71. The distance between the tooth surfaces themselves, at point 7 4, equals the sum of the individual distances 2 and z", or

a (a+z") between the tooth surfaces can be determined with the above interrelation. Such lines 7 5 are shown in Fig. 17 in which the drawing plane is also identical with the plane side 68 of the rack. The lines 7 5 are lines of equal elevation or altitude of one tooth surface from the other, adjacent point of contact 69, and are found to be ellipses. Fig. 17 thereby illustrates the nature of the point contact of gears, when each gear is provided with helical teeth.

During mesh the point of contact 69 moves on the tooth surface of each gear. The point of contact is always the intersection point between the line of action 15 and a considered tooth surface. Considering for instance the tooth surface of gear 62, the path of the point of contact on said tooth surface can be readily determined with known means, and the direction of said path adjacent point 69 is found to be identical with line 70, provided that the axes 63 and 66 are disposed at right angles to each other, as is usual. Line 7 O has then two characteristics: It is the contact line between the plane rack side 68 and the tooth surface of gear 61, and it is the path of the point 69 of contact on the tooth surface of gear 62. A. further position of the point of contact is indicated at 69 in Fig. 17

Instead of the point contact illustrated in Fig. 17 for both gears having helical teeth, contact along a line or almost along a line is desired. Stock should be left on gear 62 in such a manner that tooth contact along path 70 remains untouched, that is to say the gear action along line of action 15 should be maintained intact. The said action along line 70 is not affected when modifying the tooth surace in such a manner that additional stock is left in correspondence to the ordinates (z) of any cylindrical surface, which contacts with the plane 68 along line 70. Material added along lines parallel to line 70, but not on line itself, will not affect the gradual contact along line 70, at least when kept inside of the limits hereafter determined.

\Vhen stock is left in such manner, thehelical nature of the just considered tooth surface ends. The ordinates 2 of the tooth surface, referring to the plane 68 of the drawing, are then composed of the ordinates a" of the convex involute helicoidal tooth surface contacting with plane 68 along line 71-, and of the ordinates of the added cylindrical surface, whose generatrices are parallel to line 7 0 and which can be considered as contacting along line 70 with the plane 68 of the drawmg.

Stock may be left in the above said manner up to a degree, when contact between the mat-- ing surfaces ceases to be confined to a point (69) and extends along a line, which in the considered moment passes through point 69. The said degree is attained, when stock is left in correspondence with the ordinates a of the above said cylindrical surface having a radius of curvature C and contacting with the plane 68 along line 7 0. Contact between mating tooth surfaces then takes place along line 71. The ordinates a of the ellipsoid shown in Fig. 17, illustrative of the divergence of the mating tooth surfaces adjacent point 69, are name y composed of the ordinates 2 and 2 of the two aforesaid individual cylindrical surfaces; and when the ordinates 2 are subtracted,-only the ordinates 2" remain. These latter are zero along line 71, which is therefore the line of contact.

Frequently it is desirable to effect a contact which only approximates line contact, but not quite reaches it. Such contact is known to be less sensitive to misalignment and inaccuracies, than would be a rigid line contact, while at the same time approaching the load capacity of line contact. In this case the stock left corresponds to a cylindrical surface of equal direction, but having a radius of curvature somewhat larger than C.

The tooth surface adjacent point 69 of worm gear 62, is in any case a surface, whose ordinates 2 are composed of the ordinates z and ordinates equal to or a constant fraction of ordinates 2', depending on whether full line contactor approximate line contact is desired.

For convenience, the latter ordinates will be introduced under the symbol 2 in either case, whether they are exactly equal or only approximately equal viously referred to.

The ordinates 2 of the desired tooth surface of worm gear 62 (which corresponds to C") are determined as follows: a'=z'-a, that is to say as the difference of the ordinates z and z", inasmuch as the ordinates z are added in a direction opposite to the ordinates a". F ig.- 18 shows a topographical map of the desired tooth surface adjacent point 69. The lines 77 of equal elevationz= (5-2") are found to be hyperbolas. They have been drawn in the manner heretofore described, and the different curves have elevations a corresponding t'oan arithmetic progression.

The thread of an involute hob contacts with plane 68 along a straight line 78, whose location can be determined with the known means of the art, and which is parallel to the (normal) projection of the hob axis unto the plane 68. Like the imagined involute tooth surfaces of the gears, the thread surface can be substituted adjacent point 69 by a cylindrical surface. The said cylindrical surface is now so selected in accordance with the present invention, that it fits into the desired tooth surface, in a manner, that the straight generatrices 78 of said cylindrical surface are tangent to the corresponding hyperbolas 77. The contacting straight generatrices and hyperbolas have the some elevation from the plane of the drawing, as is uite evident. The radius of curvature R of the cylindrical surface, in a plane perpendicular to line 78, can now be determined in principle with known means. A practical way of determining radit0 the ordinates a preus R is indicated hereafter. The diameter of the hob may then be computed in the manner previously explained.

I have fully reproduced the steps taken in the analysis for determining hobs or forming members, to enable those skilled in the art not only to apply my invention, but also to understand it. The actual computation is short and accurate. It does not require drawings or diagrams, but can be carried out with a small number of simple formulas, which may be derived from the steps of the disclosed analysis. For the convenience of those who want to apply my invention, a complete set of such formulas is reproduced hereafter.

I have derived the following simple formula for determining the curvature radius R (for forming the gear C), from the given curvature radii C and C, and given angles m and m" included between a line perpendicular to line 78 and lines 70, 71, see Fig. 18. It can be checked with known means of mathamtics.

sin (m +m) Here (m +m") is the angle included between the lines and 71.

It is found that radius R is reduced, the nearer line 78 comes to an asymtote 79 of the hyperbolas 77. When line 78 lies in the direction of an asymtote 79, a zero radius R results. And when line 78 is on the other side of the asymtotes 79, that is to say in the space of hyperbolas 77, then the resulting H would be negative. When R becomes zero or negative, the method cannot be carriedout with the assumed proportions of the hob or of the gears. Then either different proportions are assumed or the generation according to the present method is attributed to the other gear of the pair of worm gears. Ordinarily it is found that conditions are favorable on one member, when they are unfavorable on the other member.

Fig. 19 is a diagram illustrative of forming gear 61 in accordance with the present invention. The principles applied are the same as explained with reference to Fig. 18. \Vhen only one gear of a pair of worm gears is formed in accordance with my invention, I usually prefer to do this on the smaller gear of the pair, that is to say on the worm, while cutting the larger gear in any suitable known way, so that its tooth sides are portions of helical surfaces.

Frequently, and especially when the pair of worm gears is of large size as compared with the size of the tool or forming member, both gears ofia pair of worm gears are formed in accordance with the present invention. A novel form of tooth is thereby obtained, which is found to have a large load capacity and which is highly efiicient.

When both gears of a pair of worm gears are formed in accordance with my novel method, stock is left on either gear of-the pair in a manner that its tooth surfaces or thread surfaces protrude over helical surfaces. Intermediate radii R and It," may be computed in the manner described with reference to curvature radius R Radius R, then constitutes the curvature radius of the tool surface in the imaginary case, when the mating gear contains helical teeth; and radius R, constitutes the curvature radius of the tool surface for producing the other worm gear of the pair, in the imaginary case, when the first said gear contains helical teeth. The stock left on the teeth of the two worm gears is then divided up in an suitable proportion, such as half and half, or one quarter and three quarters, or any other desirable proportion adapted to the particular combination. Vhen the proportion is half and half, then the radii R and R are each multiplied by two, and the hobs or forming members are determined to correspond to curvature radii (QR and (2H,). When the proportion is one quarter to three quarters, the forming members are determined to correspond to curvature radii (IE and 4 II (a Generally when the proportion is l to (1 n n the forming members are made to correspond to curvature radii (nR- and tea When the forming member is a disk wheel, such as a grinding wheel, it is preferably so s'et that its line of contact (78) with lane 68 is perpendicular to the pitch line. 11 some cases a grinding wheel is however set at another angle, that is to say at an acute angle to the pitch line. Such setting may be provided especially when an asymtote (79) coincides with or almost coincides with a line perpendicular with the pitch line (72). Another alternative is to modify the design of the teeth of the pair of worm gears, especially by altering the helix angle or the presure angle. The location of the asymtotes de ends largely on such design.

hen the forming member is a hob, or broadly a threaded member, the angle 9 included between line 78 and the pitch line 72 (on plane 68), may be determined in the same manner as for a worm gear or worm, as further indicated below. It is found that the angle g is reduced with decreasing pressure angle and with increasing lead angle h of the worm or hob. For this reason it is also found very convenient to substitute a single thread hob of reduced pressure angle to a known hob representing a worm or worm gear having multiple teeth or threads. In other words instead of providing a multiple v thread hob of the shape of a mating Worm,

"plane (68) at the in accordance with customary practice, a single thread hob havingvpreferably a reuced pressure angle may be provided in accordance with the present invention. The same result may be obtained with either hob, while in the cases particularly referred to the single thread hob may be cheaper and standard.

If so desired, proportions may be so selected, that the line of contact 78 of the hob thread coincides with the line of contact 70 of the worm thread.

The formulas below correspond to the case, where the pressure angle a of the line of action 15 is the same in generation and in the mesh of a finished gear, and Where point (39 can be considered as the pitch point, or point of contact of auxiliary cylindrical pitch surfaces having radii r and 1"". It is not deemed necessary to add the specific formulas corresponding to the use of forming members of reduced pressure angle, because such additional formulas can be derived with the known means of mathematics, after the procedure has been clearly pointed out above. Moreover my invention is notconlined to gears disposed at right angles to each other, or to specific forms of teeth, and it is clearly understood that my invention is also applicable to tapered gears. The principles of the present invention are not even confined to gears with offset axes, but are also applicable to gears with intersecting axes, as will be briefly demonstrated hereafter.

The following symbols are used in the formulas below:

1- and r=radii of the cylindrical (auxiliary) pitch surfaces.

K 13 pitch radius of forming member or hob C and C=radii of curvature at the pitch point (69) of imagined involute helicoidal tooth surfaces of the pair of worm gears.

Rp=radius of curvature of the surface of the forming member or hob, at the pitch point (69).

a pressure angle of the mean line of action (15) of the pair of worm gears.

h=lead angle at the pitch point (69) of the worm gear corresponding to C.

h.,=lcad angle of hob thread at pitch point (69). The hob thread is herein supposed to have same hand as the hand of the pair of worm gears.

q, 1/.=angles included between the pitch line (72) and the projections (70, 71) of the respective axes (63, 66) to the tangential pitch point-(69). q,.=angle included between the pitch line made smaller,

. to 90 and The formulas may be used in the following When a single thread hob is used, g, is close is almost independent of the diameter of the hob.

The curvature radius R, of a hob for forming worm gear C may be determined as follows, supposing that the mating worm gear G contains. involute helicoidal tooth surfaces:

as @W Likewise the curvature radius R of a hob may be determined as follows, supposing that the mating worm gear G contains involute helicoida-l tooth surfaces: 1/ ml! 9 0 go); l q/ o q U -cos m" 0 -cos m 5 R6, sin (g +g .Vhen both worm gears of a pair are to be cut according to the present method, and both gears are provided with tooth surfaces protruding over helical surfaces, then the curvature radii R and R are split up in the manner previously described; Otherwise the above values mrve ing the radius of the hob, which is used to cut the worm gear having such protruding tooth surfaces, that is to say whose profiles in an axial plane have a changing inclination with respect to the axis.

R 6. R; C082 (Z (3082 h wherein either R or R," is to be introduced 111 place of R depending on whether worm gear C or worm gear C feed. in the manner described.

directly for determin is cut with radial After thus determining the pitch radius of the hob, it is made sure, that its lead angle checks with the assumed lead angle h introduced in Formula If the check is not close enough, a new lead angle h maybe introduced in Formula 3, and the subsequent computation is then repeated.

When instead of a hob a grinding wheel of disk form is used, or a disk mill, angle h is introduced as zero in Formula 3. Angle g then becomes 90. Moreover m'=( m=g", according to Formulas 4. Formulas 5 remain the same; and Formula 6 can be simplified to R =R sin a.

The computations made for one side of the teeth hold good for the other side too.

Preferably the radius of the forming member'is made somewhat larger than computed in the above said manner, to obtain a slight modification from line contact.

One feature of the present invention should also be particularly noted, namely that it has made possible to accurately grind both members of a pair of worm gears with grinding wheels of disk form.

Thus far my invention has been described only as applied to gears having axes offset from the axes of the mating gears. Its principles may however also be applied to gears having parallel axes and to gears having intersecting axes, and an application to bevel gears Will now be outlined with reference to the Figures 20 to 23.

In Fig. 20 the numeral 82 denotes a bevel gear, indicated by its pitch cone and having an axis 83 and an apex 84. Its conical pitch surface is tangent to the pitch plane of a crown gear, which coincides with the plane of the drawing. The bevel gear contains spiral teeth, corresponding to a normal 85,

'hich passes through a mean point 86 of the line of contact between the conical pitch surface 82 and the pitch plane of the crown gear. The center otcurvature of the pitch line is located at point 87 on normal 8:") as well as on line 88. which is perpendicular to normal 85 and which passes through apex 84. The true crown gear is of course rotatable on an axis perpendicular to its pitch lane and passing through apex 84.

In known practice. a bevel gear 82 may be cut with a face hob. which is fed about the apex 84 and which represents the basic crown gear. It thereby gradually generates the tooth surfaces of the bevel gear 82, by forming a line of the tooth surfaces in one osit-ion of feed. and by forming other lines in further positions of feed, until the tooth surfaces are wholly generated. The said type of feed is therefore appropriately called a generating roll.

In applying my invention to the gear indicated in Fig. 20, I form gear 82 as wellas its mating gear without such generating roll. A pair of coaxial and complementary face hobs 89 are provided, having cutting teeth arranged in spirals. In the illustrated 1nstance the spirals are involutes. having a base circle 90. Preferably a single thread is employed, which extends The center of the base hob is disposed on line 88, outside of normal it is noted that with this disposition, the center of curvature of the involute thread adjacent point 86 coincides with the center of curvature 87 of the pitch line of the crown gear. Moreover the thread portion immediately adjacent point 86 is displaced bodily in the same manner, whether the portion forms part of the hob, or whether it would form part of the crown gear centered at 84:.

Each gear of the pair may be cut by rotating a face hob and the blank in timed relation, and by feeding the hob along its axis in a direction towards the blank. until final cutting depth is reached. The tooth surfaces of the blank are finished in said final position.

Inasmuch as both gears of a pair of spiral bevel gears are conjugate to the complementary thread surface of the two hobs, the tooth surfaces of said gears transmituniform motion to each other and do not interfere with each other. It is also found that spiral bevel gears produced in the above said manner mesh approximately with line contact, is desired.

A further. preferred embodiment is illustrated in Fig. 21., Fig. and Fig. 23. In this embodiment, the radius of curvature of the hob thread is larger. as is dc.-'irable. and the center of curvature lies beyond line 88. while the PIOPOltlOllS of the" gear blank 82 have otherwise been maintained the same as in Fig. 20. The hob thread extends again along an involute (92). having a base circle 93. The axis of the hob 9-1 again coincides with the center of said base circle and the hob thread has a constant pitch on all tangents 85 to its base circle.

Each side of the teeth contains a line of action, which remains unchanged and corresponds to the mesh between gear 82 and the. hob thread as well as to the mesh between gear 82 and its mating bevel gear. The two lines of action are straight lines. which arc projected into normal 85 (Fig. 21) and which are inclined to the plane of the drawing at an angle equal to the normal pressure angle (a). They pass through point 86 and fulfill the kinematical conditions of mesh between pinion 82 and either the hob thread and the mating gear.

IYhen a pair of gears is produced from a pair of complementary hobs in the manner indicated in Fig. 21, they will transmit uniform motion along the above said line of action, but mesh with point contact only. The. two gears of the pair namely contact with the complementary thread surfaces or" the two hobs along dilterent lines in any one position, and contact with each other only at the point of intersection of said lines. This point is located on the above said line of action.

along an involutc 91. clrcle and axis of the lIS To obtain line contact or approximate line contact, stock must be left on the tooth surfaces of at least one gear of the pair, excepting only a narrow strip extending along the points of contact which correspond to the said line of action. This object is attained in a procedure which is in principle the same as the one described with reference to forming worm gear pairs. Instead of providing a pair of comple-' mentary face hobs preferably one face hob and one tapered hob is provided. The smaller gear 82 of the pair is cut in the described manner with said face hob 94, and the larger gear 95 (see Fig. 22) of said pair is cut with a tapered hob 96, which is suited to mesh with the gear blank 95 along the abovesaid straight lines of action. This may be accomplished, when its thread surfaces are involute helicoidal surfaces, having the same perpendicular pitch as the perpendicular pitch of hob 94, and when its projected axis 97 passes through the above said mean point 86. The location of the base circle of a complementary face hob is indicated in dotted lines 93 in Fig. 22. The apex 98 of the tapered hob 96, or the intersection point between the axis 97 and the plane of the drawing is preferably located near base circle 93. As well known, the hand of spiral is opposite on the two bevel gears 82, 95, and this accounts for the opposite position of the hobs, according to the Figures 21 and 22.

Preferably the hand of a hob thread is made equal to the hand of the gear blank, in all embodiments of my invention. For instance gear blank 82, Fig. 20 and Fig. 21, is right hand, and the involute thread of hob 89 and of hob 94 is also right hand, as indicated.

The two hobs 94 and 96 are shown in section in Fig. 23. The profiles 99 of anaxial section of the hob threads are substantially straight and. complementary, whereas the longitudinal profiles are diverging, on account of the taper of hob 96. The said longitudinal profiles would be apparent in a section along the pitch plane 100. This is in analogy with the previously described em bodiment of the present invention.

The hobs used for cutting a pair .of worm gears in the operation previously explained also contain diverging longitudinal profiles, inasmuch as both worm gears are. produced with external hobs.

Various modifications may be made in my invention without departing from its spirit, by simply applying the established knowledge of the art. For definition of its scope I rely upon the annexed claims.

lvhat I claim is:

1. The method of forming a gear having differing profiles in planes perpendicular to its axis, and having an axis offset from and angularly disposed to the axis of its mating gear, which consists in providing a rotary forming member having forming portions disposed in a surface fitting the tooth surfaces of a gear blank so as to contact along a line with a desired tooth surface, in setting said member relatively to said gear blank at an angle other than its mating gear, in rotating said forming member, in turning the gear blank on its axis, in providing feeding motion between said forming member and said gear blank to finish the tooth surfaces of said gear blank whilemeshing with. line contact with said gear blank, and in maintaining the angular relation between the axes of the forming member and of the gear blank.

2. The method of forming agear having differing profiles in planes perpendicular to its axis and constituting the smaller member of a pair of gears having angularly disposed and offset axes, which consists in providing a rotary forming member having forming port-ions disposed in a surface fitting the tooth surfaces of a gear blank so as to contact along a line with a desired tooth surface, in setting said member relatively to said gear blank atan angle other than its mating gee r, in rotating said forming member, in turning the gear blank on its axis, in providing feeding motion between said forming member and said gear blank to finish the tooth surfaces of said gear blank while meshing with line contact with said gear blank, and in maintaining the angular relation between the axes of the forming member and of the gear blank.

3. The method of forming a gear having differing profiles in planes perpendicular to its axis, and constituting one of a pair of gears having angularly disposed and offset axes, which consists in providing a rotary member having forming portions disposed in a thread, the surface of said thread fitting the tooth surfaces of a gear blank so as to contact along a line with a desired tooth surface, in setting said member relatively to said gear blank at an angle other than its mating gear, in rotating said forming member and said gear blank at a constant ratio larger than the ratio of the said pair. in providing feeding motion between said forming member and said gear blank to finish the tooth surfaces of said gear blank while meshing with line contact with said gear blank, and in maintaining the angular relation between the axes of the forming member and of the gear blank.

4. The method of forming a gear having differing profiles in planes perpendicular to its axis and constituting one of a pair of gears having angularly disposed and offset axes, which consists in providing a hob having cutting teeth disposed in a thread, the surface of said thread fitting the tooth surfaces of a gear blank so as to contact along a line with a desired tooth surface, in setting said hob relatively to said gear blank at an angle other than its mating gear, in rotating said hob and said gear blank at a constant ratio larger than the ratio of said pair of gears, in providing feeding motion between said hob and said gear blank in a direction to approach the axes of said hob and said gear blank towards each other, and in discontinuing feed in said direction While said hob is in contact with the finished tooth sides of the gear blank.

5. The method of forming aworm having different profiles in planes perpendicular to its axis, which consists in providing a hob having cutting teeth disposed in a thread, the surface of said thread fitting the tooth surfaces of a worm blank so as to contact along a line with a desired tooth surface, in rotating said hob and said worm blank at a constant ratio on angularly disposed and offset axes, in providing feeding motion between said hob and said worm'blank in a direction to approach the axes of said hob and said worm blank, and in discontinuing feed in said direction while said hob is in contact with the finished worm.

6. The method of forming a worm having differing profiles in planes perpendicular to its axis, which consists in providing a hob having cutting teeth disposed in a thread, the surface of said thread fitting the tooth surfaces of a worm blank so as to contact along a. line with a desired tooth surface, in rotating said hob and said worm blank at a constant ratio on angularly disposed and offsct axes, in providing feeding motion between said hob and said worm blank to approach the axes of said hob and said worm blank, and in maintaining the angular relation between the axes.

T. The method of forming a gear having differing profiles in planes perpendicular to its axis, which consists in providing a hob having cutting teeth disposed in a thread, the surface of said thread fitting the tooth surfaces of a gear blank so as to contact along a line with a desired tooth surface, in rotating said hob and said gear blank at a constant ratio on angularly disposed and offset axes, in providing feeding motion to approach the axes of said hob and said gear blank, and in gradually slowing up the feeding motion with increasing depth of cut.

8. The method of forming a gear having differing profiles in planes perpendicular to its axis and having said angularly disposed to the axis of its mating gear, which consists in providing a hob having cutting teeth disposed in a thread, in setting said hob relatively to a gear blank at an angle other than its mating gear, in rotating said hob and said gear blank at a constant ratio,

in providing feeding notion to approach the axes of said hob and said gear blank, and in gradually slowing up the feeding motion with increasing depth of cut.

t). The method of ,forming a gear having differing profiles in planes perpcinlicnlar to its axis, said gear being the smaller gear of a pair, which consists in providing a forming member having forming portions disposed in a thread, in rotating said forming member and a gear blank in timed relation to each other on angularly disposed and ofi'set axes to turn said forming member at a higher rate than said gear blank, in providing feeding motion in a directon to approach said forming member and said gear blank, and in discontinuing feed in said direction while the forming member is in mesh with the finished gear blank.

10. The method of forming a gear having differing profiles in planes perpemlicular to its axis, said gear being the smaller gear of a pair, which consists in providing a hob having cutting teeth disposed in a thread, in settingsaid hob relatively to a gear blank at an angle other than its mating gear, in rotating said hob and said gear blank in timed relation to each other in a manner to turn said hob at a higher rate than said gear blank, in providing feeding motion between said hob and said gear blank in a direction inclined to the. axis of said gear blank while maintaining the angular relation between the axes of the hob; and the gear blank.

11. The method of forming a gear having differing profiles in planes perpendicular to its axis, said gear being the smaller gear of a pair, which consists in providing :1 bob havingcutting teeth disposed in a thread, in setting said hob relatively to a gear blank. at an angle other than its mating gear, in rotating said hob and said gear blank in timed relation to each other in a manner to turn said hob at a higher rate than said gear blank, in providing feeding motion between said hob and said gear blank in a direction to approach said hob and said'gear blank, in maintaining the axes of said hob and said gear blank in constant angular relation, and in discontinuing feed in said direction while said hob is in meshing contact with the finished gear. 7 i

12. The method of forming one of a pair of worm gears, which consists in providing a rotary forming member having a pressure angle other than the pressure angle of said gear pair at a mean point of mesh. said mean point of mesh being the intersection point of the surface of action with the shortest connection line between the axes of said gear pair, in rotating said forming member and a gear blank on angularlv disposed and offset axes, and in providing feeding motion between said forming inembe' and said gear blank to accurately finish said gear blank.

13. The method of forming one of a pair of worm gears, which consists in providing a rotary forming member having forming portions disposed in a helicoidal thread surface of different pressure angle than the pres sure angle of said gear pair at a mean point Inn ill)

of mesh, said mean point of mesh being the intersection point of the surface of action with the shortest connection line between the axes of said gear pair, in rotating said forming member and a gear blank on angularly disposed and offset axes, and in providing feeding motion between said forming member and said gear blank along a straight line inclined to the axis of said gear blank to effect engagement between said helicoidal thread surface and the tooth surfaces of said worm gear.

14. The method of forming one of a pair of worm gears, which consists in hob having a number of threads smaller than the number of teeth of the other worm gear of said pair and having a pressure angle other than the mean pressure angle of said pair, in rotating said hob and a gear blank in timed relation on angularly disposed and offset axes, and in providing feeding motion between said hob and said gear blank along a straight line inclined to the axis of said gear blank.

15. The method offorming one of a pair of Worm gears having multiple teeth, which consists in providing a hob having a single thread and a reduced pressure angle, in retating said hob and a gear blank in timed relation on angularly disposed and offset axes, and in providing feeding motion between said hob and said gear blank along a straight line inclined to the axis of said gear blank.

16. The method of forming a gear having differing profiles in planes perpendicular to its axis, which consists in providing a hob having a pressure angle other than the mean pressure angle of said gear, in rotating said hob and a gear blank in timed relation on angularly disposed and offset axes, viding feeding motion between said hob and said gear blank along a straight line inclined to the axis of said gear blank.

17. The method of forming a pair of gears having angularly disposed and offset axes, which consists in forming each gear of said pair by providing a rotary forming member, said forming member containing an involute helcoidal thread surface, in setting said member relatively to a gear blank at an angle other'than its mating gear, in rotating said forming member, in turning the gear blank on its axis, in providing feeding motion between said forming member and said gear blank along a. straight line inclined to the axis of said gear blank, and'in maintaining a constant angular relation between said axes.

18. The method of forming a pair of gears having angularly disposed and offset axes, which consists in forming each gear of said )air with a rotary forming member having l orming portions disposed in a surface of convex profile in a section at right angles to its axis, in rotating said forming member, in

providing a and in pro- -h turning a gear blank on its said forming member and the axis of said gear blank being angularly disposed to and offset from each other and in providing feeding motion between said forming member and said gear blank in a direction along a straight line inclined to the axis of the gear blank.

19. The method of forming a pair of gears having angularly disposed axes, at least one of said gears having differing profiles in sections perpendicular to its axis, which consists in providing a pair of forming members having diverging longitudinal profiles, and in forming each gear of said pair by rotating one of said forming with a gear blank, by turning said gear blank onits axis, the axis of said gear blank and the axis of said forming member being angularly disposed to and offset from each other, and by providing linear feeding motion in a direction at an angle to the axis of said gear blank.

20. The method of forming a pair of gears axis, the axis of having angularly disposed axes, at least one of which having differing profiles in planes perpendicular to its axis, which consists in providing a pair of forming members having substantially complementary profiles in axial planes and having diverging longitudinal profiles, and in forming each gear of said pair by rotating one of said forming members in engagement with a gear blank, by turning said gear blank on its axis, the axis of said gear blank and the axis of said forming member being angularly disposed to and offset from each other, feeding motion in a direction at an the axis of the gear blank.

21. The method of forming a pair of gears aving angularly disposed axes, at least one of Which having differing profiles in planes perpendicular to its axis, which consists in providing a pair of forming members of diverging longitudinal profiles, said forming members having forming portions disposed in helicoidal surfaces, in mounting said forming members adjacent a pair of gear blanks respectively on axes offset from and angularly disposed to the axes of the respective gear blanks, and in forming said pair of gears conjugate'to the two forming members respectively, by rotating said forming members and the respective gear blanks on their axes in engagement with each other, so that a thread surface of said forming members engages the entire conjugate tooth surfaces of the respective gear blanks in a single bodily position.

22. The method of forming a pair of gears having angularly disposed axes, at least one of which having differing profiles in planes perpendicular to its axis, which consists in providing a pair of forming members of diverging longitudinal profiles, said forming angle to members in engagement 7 and by providing linear members having forming portions disposed in helicoidal surfaces having substantially complementary profiles in an axial (plane, in mounting said formingmembers a jacent a pair of gear blanks respectively on axes offset from and angularly disposed to the axes of the respective gear blanks, and in forming said pair of gears conjugate to the two forming members respectively, by rotating said forming members and the respective gear blanks on their axes in engagement with each other, so that a thread surface of said forming members engages the entire'conjugate tooth surfaces of the respective gear blanks m a single bodily position.

23. The method of formin a pair of gears, at least one of which having difierin profiles in planes perpendicular to its axis, w iich consists in providing a pair of forming members having forming portions disposed in sin le threads of diverging longitudinal proti es, and in forming said pair of gears with said pair of forming members by rotating a forming member and gear blank in timed relation on angularly disposed and otiset axes, and by prbviding feeding motion between said forming member and said gear blank along a straight line to ap roach sa d forming member and said gear lank while maintaining the axes of said forming member and said gear blank in constant angular relation.

24. The method of forming a pair of gears having angularly' disposed axes, at least one of which having differing profiles 1n planes perpendicular to its axis, which consists in providing a pair of forming members having forming portions disposed in involute helicoidal thread surfaces, the hand of said thread surfaces being equal to the hand of the respective gears, in mounting said forming members adjacent two gear blanks on axes ottset from and angularly disposed to the axes of the respective gear blanks, and in form ng said air of gears with said pair of forming mem ers by rotating a forming member and a gear blank in timed relation in a manner to turn said forming member at a higher rate, and by providing feeding motion between said forming member and said gear blank in a direction at an angle to the axis of the gear blank while maintaining a constant angular relation between the axes of said forming member and of said gear blank.

2-5. The method of forming a pair of gears having angularly disposed axes, at least one of which having diifering profiles in planes at right angles to its axis, which consists in providing a pair of forming members having forming portions disposed in single threads, the hand of said threads being equal to the hand of the respective gears, in mounting said forming members adjacent two gear blanks respectively on axes offset from and angularly disposed to the axes of the respective gear blanks, and in forming said pair of gears with said pair of formin members b rotating a forming member an a gear blan t on their axes in engagement with each other, and by providing feeding motion between said forming member and said gear blank in a direction at an angle to the axes of the forming member and of the gear blank, while maintaining a constant angular relation between the axes of the forming member and of the ar blank.

26. he method of forming a pair of gears ha-vin angularly disposed axes, at least one of which having'differing profiles in planes at right angles to its axis, which consists in providing a air of hobs having cutting teeth disposed in elicoidal thread surfaces, said hobs having diverging longitudinal profiles, in mounting each hob adjacent agear blank on an axis angularly disposed to and offset from the axis of said gear blank. the angle included between the axes of the hob and of said gear blank differing more than ten degrees (10) from the angle between the axes of the gear pair. in rotating each hob and the respective gear blank on their axes in timed relation to each other, in providing feeding motion between said hob and said-gear blank in a direction to approach said hob and said gear blank, and in discontinuing feed in said direction while said hob is in engagement with the finished gear blank.

27. The method of forming a pair of gears having angularly disposed axes, at least one of which having differing profiles in planes at right angles to its axis, which consists in providing a pair of hobs having cutting teeth disposed in single threads, the normals to the side surfaces of said threads having a constant inclination to the axis of the respective hob and being an equal distance offset from said axis, and in forming said pair of gears with said pair of hobs, by rotating :1 hob and a gear blank in timed relation with each other on angularly disposed and offset axes, by providing feeding motion along a straight line between said i a direction to approach said hob and said gear blank. in maintaining a constant angular relation bet-ween the axes of said hob and said gear blank, and in discontinuing feed in said direction while said hob is in engagement with the finished tooth surfaces of the gear blank.

28. The method of forming a pair of worm gears, which consists in providing a pair of external hobs having cutting teeth disposed in helicoidal threads, in mounting said li'obs adjacent two gear blanks respectively on axes offset from and singularly disposed to the axes of the respective gear blanks,the angle between the direct-ions of the axes of a hob and of the respective gear blank differing more than ten degrees (10) fronrihe angle between the axes of said pair of worth gears.

hob and said gear blank in and in forming said pair of worm gears with said pair of hobs by rotating a hob andleast one of said hobs having the tips of said cutting edges disposed in a. surface of revolution of curved profile, in mounting said hobs adjacent two gear blanks respectively onaxes offset from and angularly disposed to the axes of the respective gearblanks, and in cutting said pair of worm gears with said air of hobs, by rotating a hob and a gear ilank in timed relation, by providing feeding motion between said hob and said ear blank in a direction to approach said hob and said gear blank, and by discontinuing feed in said direction while said hob is in mesh at full depth with the finished-tooth surfaces of the gear blank.

30. The method of forming a pair of conjugate gears having angularly dis osed axes, at least one of which having differing profiles in planes perpendicular to its axis, which consists in providing a pair of rotary forming members conjugate to said gears and differing from both gears of said pair, corresponding forming surfaces of said forming members having diverging longitudinal profiles, in mounting said forming members and the respective, gears on angularly di osed and offset axes, and in rotating saic forming members in engagement with the respective rotating gears while maintaining a constant angular relation between the axes of said forming members and the axes of the respective gears.

Y 31. The method of forming a pair of conjugate gears havin angnlarly disposed axes, at least one of whic having differing profiles 'in planes at right angles to its axis, which consists in providing 'a pair of rotary forming members conjugate to said gears and differing profiles in planes at right angles to its axis, which consists in providing-a pair of forming members having forming portions disposed in surfaces fitting the tooth surfaces of the respective gears so as to contact along a line. with a desired tooth surface, in rotating said gears on their axes, in rotating said forn'iing members in engagement with the respective gears on axes angularly disposed to and offset from the axes of the respective gears, and in providing feeding motion along a straight line between said forming members and the respective gears. I

33. The method of forming a pair of conj agate gears having angularly disposed and oifsetaxes, at least one of which having differing profiles in planes at right angles to its axis, which consists in providing a pair of hobs having cutting edges disposed in threads 3 fitting the tooth surfaces of the respective gears so as to contact along a line with a desired tooth surface. in rotating said gears on their axes, in rotating said hobs in engagement with the respective gears on axes angularly disposed to and offset from the axes of the respective gears, and in providing feeding motion between said-hobs and the respective gears ina manner to approach the axes of said hobs and of said gears relatively to each other.

34. The method of forming one of a pair of worm gears, which consists in providing a rotary forming member having forming portions disposed in an involute helicoidal thread surface of different pressure angle than the pressure angle of said gear .pair at a mean point of mesh, said mean point of mesh being the intersection point of the surface of action with the shortest connectionlinc between the axes of said gear pair, in rotating said forming member and a gear blank on angularly disposed and offset axes, and in providing feeding motion between said forming member and said gear blank along a straight line inclined to .the axis of said gear blank to effect engagement between said helicoidal thread surface and the tooth surfaces of said worm gear. 1

35. The method of forming a worm, which consists in providing a forming member having forming portions disposed in'a heliooidal farm from both gears of Said P thread surface, in mounting said forming said orming'members having pitch surfaces which are convexin peripheral direction, in mounting said forming members and therespective gears on angularly disposed and off setmxes, and in rotating said forming mom hers in engagement with the respective rotating gears while maintaining a constant angular relation between the axes of said forming members and the axes of the respective gears.

32. The method of forming a pair of conjugate gear. having angularly disposed and offset axes, at least one of which having difleast one side of the teeth of said worm blank in a single position of the rotating bodies of the worm blank and said forn'iing member.

36. The method of forming the smaller gear of a gear pair having angularly dis posed axes, which consists in providing a blank, and in discontinuing feed in said direction while said hob is in mesh at full depth with the finished tooth surfaces of the gear blank.

ERNEST TVILDHABER.

forming member having forming portions 7 4 axes of said gear pair, and in forming said gear conjugate to said helicoidal thread surface, by rotating said gear blank and said forming member on their respective axes in engagement with each other so that said thread surface engages the entire tooth surfaces of at least one side of the teeth of said gear blank in a single position of the rotating bodies of the gear blank and said forming member.

37. The method of forming a gear of a pair'of gears having angularly disposed axes, which consists in providing a forming member having forming portions disposed in an involute helicodial thread surface, said thread surface having a pressure angle' differing from the pressure angle of said gear pair at a mean point of mesh, in mounting said forming member adjacent a gear blank on an axis offset from and angularly disposed to the axis of said gear blank, the angle included between the directions of said two axes differing at least ten degrees (10) from the angle between the axes of said gear pair, and in forming said gear conjugate to said 40 helicoi dal thread surface, by rotating said gear blank and said forming member on their respective axes in engagement with each other so that said thread surface engages the entire tooth surfaces of at least one side of the teeth of said gear blank in a single position of the rotating bodies of the gear blank and said forming member;

38. The method of forming one of a pair of Worm gears, which consists in providing 59 a hub having cutting teeth projecting outwardly from a round body portion, said cutting teeth having tips disposed in a surface of revolution of curved profile and containing cutting edges disposed in a helicoidal surface, in mounting said hob adjacent a gear blank on an axis offset from and angularly disposed to the axis of said gear blank, the angle included between the directions of said axes differing more than ten degrees (10) 60 from the angle between the axes of said pair of gears, in rotating said hob and said gear blank in timed relation in engagement'with each other, in providing feeding motiori between said hob and said gear blank in a direction to approach said hob and said gear 

